Deep Learning-Based Reconstruction of Particle Beam Energy Spectra from Attenuation Curve Data

This paper introduces a deep learning-based methodology for reconstructing particle beam energy spectra from experimental attenuation curves. This task involves solving a classic ill-posed inverse problem for a Fredholm integral equation of the first kind. Unlike traditional Arsenin–Tikhonov regularization, the proposed framework utilizes two coupled neural networks for spectrum approximation and adaptive kernel correction. This approach explicitly accounts for measurement uncertainties in the experimental data. As a mesh-free technique, it operates directly on raw sparse experimental datasets without preprocessing. Validation using data from subnanosecond electron beams in gas-filled and vacuum diodes demonstrates that the method successfully resolves non-trivial two-peak spectral structures. In particular, it reliably identifies populations of “anomalous” high-energy electrons that are often obscured by classical regularization artifacts.